To extend the valid range of the hopping parameter expansion,
we thus revisit the effective theory of Refs. [1]}, [2]}, [3]}
which incorporates the next-to-leading effect into the leading-order calculation,
and extend it to higher-orders of the hopping parameter expansion.
We also discussed that the effect of Wilson loop terms can be represented by a shift of coupling parameters in the gauge action.
The effective theory is based on the strong correlation between the leading-order Polyakov loop and next-to-leading bend Polyakov loops.
By a Monte-Carlo simulation, we showed that the strong correlation holds also for higher-order Polyakov-type loops.
We thus extended the effective theory to include higher-order terms of the hopping parameter expansion,
and determined the coefficients needed in the effective theory.
Using the effective theory, we discussed that the truncation error of the hopping parameter expansion is well under control for \(\kappa \mathrel {\unknown. {<}}{\sim }\) 0.125\( when the higher-order effect is incorporated into the effective theory up to sufficiently high orders.We evaluated the higher-order correction of the critical point \) c\( for \) Nt=6\(.We also derived formulae for the critical point with general number of flavors at zero and finite densities.\)
